Finite Group Databasebeta

← #29680744

Subgroups — #29680744

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C2×#16839805C211yesmaximal, characteristic, complemented
3C2×#16839725C211yesmaximal, characteristic
4#21448198C211yesmaximal, characteristic
5#21446010C214yesmaximal, complemented
6#21446005C214yesmaximal, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C2×C61×#428C211yesmaximal, characteristic, complemented
10C23×C4×D61—21no—
11C23×C4×D61C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×#11758196—21no—
16C2×C4⋊C4×D61—24no—
17C2×C4⋊C4×D61—24no—
18C2×C4⋊C4×D61—22no—
19C2×C4⋊C4×D61C2214yes—
20#16839805—24no—
21C2×#14519710C2212yescomplemented
22C2×#14519708C2212yescomplemented
23#16839725C418yescomplemented
24C2×#14519689C2214yes—
25#16839623—21no—
26#16839619C2212yes—
27#16839619C2212yes—
28#16839617—21no—
29C4×#11758198—22no—
30#16839525C2214yescomplemented
31#16839524C2214yescomplemented
32C22×M16×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#428—24no—
36C2×#107×C61C2212yescomplemented
37C2×#106×C61C2212yescomplemented
38C24×D61D411yescharacteristic
39C23×Dic61—21no—
40C23×Dic61D411yescharacteristic
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—28no—
46C22×C4×D61—24no—
47C22×C4×D61—48no—
48C22×C4×D61C2×C414yes—
49C22×C4×D61D412yes—
50C22×C4×D61Q814yes—
51C22×C4×D61C2312yes—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C2×#11758194C2×C414yescomplemented
55C22×#92418C2312yes—
56#14519736—22no—
57#14519736—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61D414yes—
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—28no—
64C2×#11758196—24no—
65C4⋊C4×D61—216no—
66C4⋊C4×D61—28no—
67C4⋊C4×D61—28no—
68C2×C22⋊Dic61—21no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—21no—
72#14519710—24no—
73#14519708—24no—
74#14519689C2×C4116yescomplemented
75#14519684—22no—
76#14519684—22no—
77C23×C4×C61—21no—
78C23×C4×C61C2311yescharacteristic
79C2×M16×C61—22no—
80C2×M16×C61—22no—
81C2×M16×C61C2×C414yescomplemented
82C22×C8×C61C2312yes—
83#117×C61—22no—
84#117×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61D414yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2312yes—
91C2×C22⋊C4×C61—21no—
92C2×C42×C61—21no—
93#107×C61—24no—
94#106×C61—24no—
95C24×C61C2×D411yescharacteristic
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—24no—
99C23×D61—24no—
100C23×D61C4⋊C414yes—
101C23×D61C2×D412yes—
102C22×C4×C61—24no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—44no—
107C22×C4×C61C22×C412yes—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×Q812yes—
112C22×C4×C61#4112yes—
113C22×C4×C61#4112yes—
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—24no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—44no—
121C22×Dic61C4⋊C414yes—
122C22×Dic61C2×D412yes—
123C2×C4×D61—28no—
124C2×C4×D61—28no—
125C2×C4×D61—24no—
126C2×C4×D61—28no—
127C2×C4×D61—28no—
128C2×C4×D61—24no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—216no—
134C2×C4×D61—432no—
135C2×C4×D61—48no—
136C2×C4×D61—48no—
137C2×C4×D61C4⋊C418yes—
138C2×C4×D61C22×C414yes—
139C2×C4×D61C2×Q814yes—
140M16×C61—28no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61C22×C414yescomplemented
144C4⋊C4×C61—24no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D414yes—
151C22⋊C4×C61—42no—
152C42×C61—42no—
153C22⋊Dic61—42no—
154#11758198—44no—
155#11758198—44no—
156C4⋊Dic61—24no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C2×D414yes—
161#11758196—216no—
162C4×Dic61—42no—
163#11758194—28no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418C22×C414yescomplemented
167C23×C61—22no—
168C23×C61—22no—
169C23×C61C2×C4⋊C412yes—
170C23×C61#12012yes—
171C23×C61#12312yes—
172C23×C61C22×D411yescharacteristic
173C22×D61—28no—
174C22×D61—216no—
175C22×D61—28no—
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—28no—
181C22×D61—216no—
182C22×D61C2×C4⋊C414yes—
183C22×D61AGL(1,ℤ/8)14yes—
184C22×D61#13714yes—
185C2×C4×C61—24no—
186C2×C4×C61—48no—
187C2×C4×C61—44no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61C2×C4⋊C412yes—
193C2×C4×C61C4×D414yes—
194C2×C4×C61C4×D414yes—
195C2×C4×C61#12012yes—
196C2×C4×C61C22⋊Q814yes—
197C2×C4×C61C22⋊Q814yes—
198C2×C4×C61#12312yes—
199C2×C4×C61C23×C411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×Q811yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—28no—
208C2×Dic61—24no—
209C2×Dic61—48no—
210C2×Dic61—44no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61C2×C4⋊C414yes—
215C4×D61—24no—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—24no—
219C4×D61—216no—
220C4×D61—432no—
221C8×C61—24no—
222#92418—24no—
223#20844—611nomaximal, Sylow
224C22×C61—24no—
225C22×C61—24no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61#38514yes—
229C22×C61C22×C4⋊C411yescharacteristic
230C22×C61C2×#12011yescharacteristic
231C22×C61C2×#12311yescharacteristic
232C22×C61C2×AGL(1,ℤ/8)12yes—
233C22×C61C2×#13712yes—
234C2×D61—232no—
235C2×D61—24no—
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—24no—
239C2×D61—24no—
240C2×D61—24no—
241C2×D61#42818yescomplemented
242C4×C61—44no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61#38514yes—
246C4×C61C2×C4×D411yescharacteristic
247C4×C61C2×C4×D411yescharacteristic
248C4×C61C2×C22⋊Q811yescharacteristic
249C4×C61C2×C22⋊Q811yescharacteristic
250Dic61—24no—
251Dic61—44no—
252Dic61—82no—
253C23×C4⋊C4—611no—
254C2×#428—611no—
255C2×#428D6111yescharacteristic, complemented
256C2×#407—611no—
257#2280—611no—
258#1853—614no—
259#1852—614no—
260#1736—611no—
261#1736—611no—
262C2×C61—24no—
263C2×C61C2×#38511yescharacteristic
264C2×C61C2×#42812yes—
265C2×C61#300311yescharacteristic
266C2×C61#300411yescharacteristic
267C2×C61#309212yes—
268D61—28no—
269C24×C4—611no—
270C24×C4—1221no—
271C22×M16—611no—
272C22×M16C2×D6111yescharacteristic
273C2×#117—611no—
274C2×#117C2×D6111yescharacteristic
275C22×C4⋊C4—611no—
276C22×C4⋊C4—614no—
277C22×C4⋊C4—1224no—
278C22×C4⋊C4—1222no—
279C22×C4⋊C4—1221no—
280C22×C4⋊C4—1224no—
281C22×C4⋊C4C2×D6111yescharacteristic
282#428—24no—
283#428—1224no—
284C2×#107—612no—
285C2×#107C2×D6112yescomplemented
286C2×#106—612no—
287C2×#106C2×D6112yescomplemented
288#407—618no—
289C2×C22⋊C8—614no—
290#381—1221no—
291#380—612no—
292#380—612no—
293#379—1221no—
294C4×C22⋊C4—1222no—
295#340—614no—
296#340—614no—
297C61#2084411yescharacteristic, complemented, Sylow
298C25—611no—
299C23×C4—21no—
300C23×C4—611no—
301C23×C4—614no—
302C23×C4—612no—
303C23×C4—614no—
304C23×C4—612no—
305C23×C4—1224no—
306C23×C4—1228no—
307C23×C4—1222no—
308C23×C4—1222no—
309C23×C4—1222no—
310C23×C4—1222no—
311C23×C4—1221no—
312C23×C4—2448no—
313C23×C4C22×D6111yescharacteristic
314C2×M16—22no—
315C2×M16—22no—
316C2×M16—614no—
317C2×M16—1222no—
318C2×M16—1222no—
319C2×M16C4×D6114yescomplemented
320C22×C8—612no—
321C22×C8C22×D6112yes—
322#117—22no—
323#117—22no—
324#117—1222no—
325#117—1222no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—614no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1228no—
332C2×C4⋊C4—12216no—
333C2×C4⋊C4—1224no—
334C2×C4⋊C4—1228no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1228no—
338C2×C4⋊C4C22⋊D6114yes—
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C4⋊C4C22×D6111yescharacteristic
341C2×C4⋊C4C22×D6112yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1221no—
346C2×C42—21no—
347C2×C42—1221no—
348#107—24no—
349#107—1224no—
350#106—24no—
351#106—1224no—
352C22⋊C8—6116no—
353#95—1222no—
354#95—1222no—
355C24—612no—
356C24—614no—
357C24—1224no—
358C24—1224no—
359C24—1222no—
360C24—1222no—
361C24D4×D6111yescharacteristic
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—44no—
367C22×C4—612no—
368C22×C4—614no—
369C22×C4—614no—
370C22×C4—614no—
371C22×C4—618no—
372C22×C4—12216no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1228no—
380C22×C4—1228no—
381C22×C4—1224no—
382C22×C4—1228no—
383C22×C4—1228no—
384C22×C4—1224no—
385C22×C4—1222no—
386C22×C4—1224no—
387C22×C4—1222no—
388C22×C4—2448no—
389C22×C4—24432no—
390C22×C4—2448no—
391C22×C4—2444no—
392C22×C4C2×C4×D6112yes—
393C22×C4#1175822212yes—
394C22×C4D4×D6112yes—
395C22×C4D4×D6111yescharacteristic
396C22×C4Q8×D6112yes—
397C22×C4#1175822612yes—
398C22×C4C2×C22⋊D6112yes—
399C22×C4C23×D6111yescharacteristic
400M16—28no—
401M16—1228no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—614no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8C2×C4×D6114yescomplemented
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—614no—
415C4⋊C4—12216no—
416C4⋊C4—1222no—
417C4⋊C4—1224no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4C2×C22⋊D6114yes—
421C22⋊C4—42no—
422C22⋊C4—2444no—
423C22⋊C4—2444no—
424C22⋊C4—2442no—
425C42—42no—
426C42—2442no—
427C23—22no—
428C23—22no—
429C23—614no—
430C23—614no—
431C23—614no—
432C23—12216no—
433C23—1228no—
434C23—12216no—
435C23—1228no—
436C23—1222no—
437C23—1222no—
438C23—1222no—
439C23—1222no—
440C23—1228no—
441C23C4⋊C4×D6112yes—
442C23#1451974512yes—
443C23#1451976212yes—
444C23C2×D4×D6111yescharacteristic
445C2×C4—24no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—614no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1228no—
456C2×C4—1224no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—1224no—
460C2×C4—12216no—
461C2×C4—1224no—
462C2×C4—24432no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2448no—
467C2×C4—2444no—
468C2×C4C4⋊C4×D6112yes—
469C2×C4C4×C22⋊D6114yes—
470C2×C4#1451973914yes—
471C2×C4#1451974512yes—
472C2×C4#1451975314yes—
473C2×C4#1451975514yes—
474C2×C4#1451976212yes—
475C2×C4C22×C4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C22×C22⋊D6111yescharacteristic
478C2×C4C2×Q8×D6111yescharacteristic
479C2×C4C2×#1175822211yescharacteristic
480C2×C4C2×#1175822611yescharacteristic
481C8—24no—
482C8—1224no—
483C22—24no—
484C22—24no—
485C22—22no—
486C22—22no—
487C22—618no—
488C22—12232no—
489C22—1224no—
490C22—1224no—
491C22—1224no—
492C22—1224no—
493C22—1224no—
494C22—1224no—
495C22#1683963314yes—
496C22C2×C4⋊C4×D6111yescharacteristic
497C22C2×#1451974511yescharacteristic
498C22C2×#1451976211yescharacteristic
499C22AGL(1,ℤ/8)×D6112yes—
500C22#137×D6112yes—
501C4—44no—
502C4—81no—
503C4—81no—
504C4—1224no—
505C4—4882no—
506C4—2444no—
507C4#1683963314yes—
508C4C2×C4×C22⋊D6111yescharacteristic
509C4C2×#1451973911yescharacteristic
510C4C2×#1451975311yescharacteristic
511C4C2×#1451975511yescharacteristic
512C2—24no—
513C2—1228no—
514C2C2×#1683963311yescharacteristic
515C2#428×D6112yes—
516C2#2145504411yescharacteristic
517C2#2145505111yescharacteristic
518C2#2145574512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC2×#428D61
hypocenterC61#20844
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839805C2semidirect
5. #21446010C2semidirect
6. #21446005C2semidirect
7. #21445539C2semidirect
8. #21445538C2semidirect
9. C2×C61×#428C2semidirect
21. C2×#14519710C22semidirect
22. C2×#14519708C22semidirect
23. #16839725C4semidirect
30. #16839525C22semidirect
31. #16839524C22semidirect
36. C2×#107×C61C22semidirect
37. C2×#106×C61C22semidirect
54. C2×#11758194C2×C4semidirect
74. #14519689C2×C4semidirect
81. C2×M16×C61C2×C4semidirect
143. C2×C8×C61C22×C4semidirect
166. C2×#92418C22×C4semidirect
241. C2×D61#428semidirect
255. C2×#428D61semidirect
285. C2×#107C2×D61semidirect
287. C2×#106C2×D61semidirect
297. C61#20844semidirect
319. C2×M16C4×D61semidirect
407. C2×C8C2×C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C4⋊C4×D612
C2×#168398052
C2×#168397252
#214481982
#214460102
#214460052
#214455392
#214455382
C2×C61×#4282
#2084461